Construction of infinite dimensional interacting diffusion processes through Dirichlet forms

1996 ◽  
Vol 106 (2) ◽  
pp. 265-297 ◽  
Author(s):  
Minoru W. Yoshida
Keyword(s):  
Diffusion Processes ◽  
Dirichlet Forms ◽  
Infinite Dimensional ◽  
Interacting Diffusion

DIRICHLET FORMS IN TERMS OF WHITE NOISE ANALYSIS II: CLOSABILITY AND DIFFUSION PROCESSES

1989 ◽  
Vol 01 (02n03) ◽  
pp. 313-323 ◽  
Author(s):  
S. ALBEVERIO ◽  
T. HIDA ◽  
J. POTTHOFF ◽  
M. RÖCKNER ◽  
L. STREIT
Keyword(s):  
White Noise ◽  
Diffusion Processes ◽  
Noise Analysis ◽  
Dirichlet Forms ◽  
White Noise Analysis ◽  
Topological Vector Spaces ◽  
Infinite Dimensional ◽  
Dimensional Diffusion ◽  
White Noise Functionals ◽  
And Diffusion

It is shown that infinite dimensional Dirichlet forms as previously constructed in terms of (generalized) white noise functionals fit into the general framework of classical Dirichlet forms on topological vector spaces. This entails that all results obtained there are applicable. Admissible functionals give rise to infinite dimensional diffusion processes.

Effective resistances for harmonic structures on p.c.f. self-similar sets

1994 ◽  
Vol 115 (2) ◽  
pp. 291-303 ◽  
Author(s):  
Jun Kigami
Keyword(s):  
Diffusion Processes ◽  
Difference Operator ◽  
Scaling Limit ◽  
Sierpinski Gasket ◽  
Dirichlet Forms ◽  
Sierpiński Gasket ◽  
Know How ◽  
Analysis On Fractals ◽  
Topological Description ◽  
Self Similar

In mathematics, analysis on fractals was originated by the works of Kusuoka [17] and Goldstein[8]. They constructed the ‘Brownian motion on the Sierpinski gasket’ as a scaling limit of random walks on the pre-gaskets. Since then, analytical structures such as diffusion processes, Laplacians and Dirichlet forms on self-similar sets have been studied from both probabilistic and analytical viewpoints by many authors, see [4], [20], [10], [22] and [7]. As far as finitely ramified fractals, represented by the Sierpinski gasket, are concerned, we now know how to construct analytical structures on them due to the results in [20], [18] and [11]. In particular, for the nested fractals introduced by Lindstrøm [20], one can study detailed features of analytical structures such as the spectral dimensions and various exponents of heat kernels by virtue of the strong symmetry of nested fractals, cf. [6] and [15]. Furthermore in [11], Kigami proposed a notion of post critically finite (p.c.f. for short) self-similar sets, which was a pure topological description of finitely ramified self-similar sets. Also it was shown that we can construct Dirichlet forms and Laplacians on a p.c.f. self-similar set if there exists a difference operator that is invariant under a kind of renormalization. This invariant difference operator was called a harmonic structure. In Section 2, we will give a review of the results in [11].

Sign in / Sign up

Export Citation Format

Share Document